(* Author: Florian Haftmann, TU Muenchen *)
section \<open>Lexical order on functions\<close>
theory Fun_Lexorder
imports Main
begin
definition less_fun :: "('a::linorder \<Rightarrow> 'b::linorder) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
where
"less_fun f g \<longleftrightarrow> (\<exists>k. f k < g k \<and> (\<forall>k' < k. f k' = g k'))"
lemma less_funI:
assumes "\<exists>k. f k < g k \<and> (\<forall>k' < k. f k' = g k')"
shows "less_fun f g"
using assms by (simp add: less_fun_def)
lemma less_funE:
assumes "less_fun f g"
obtains k where "f k < g k" and "\<And>k'. k' < k \<Longrightarrow> f k' = g k'"
using assms unfolding less_fun_def by blast
lemma less_fun_asym:
assumes "less_fun f g"
shows "\<not> less_fun g f"
proof
from assms obtain k1 where k1: "f k1 < g k1" "k' < k1 \<Longrightarrow> f k' = g k'" for k'
by (blast elim!: less_funE)
assume "less_fun g f" then obtain k2 where k2: "g k2 < f k2" "k' < k2 \<Longrightarrow> g k' = f k'" for k'
by (blast elim!: less_funE)
show False proof (cases k1 k2 rule: linorder_cases)
case equal with k1 k2 show False by simp
next
case less with k2 have "g k1 = f k1" by simp
with k1 show False by simp
next
case greater with k1 have "f k2 = g k2" by simp
with k2 show False by simp
qed
qed
lemma less_fun_irrefl:
"\<not> less_fun f f"
proof
assume "less_fun f f"
then obtain k where k: "f k < f k"
by (blast elim!: less_funE)
then show False by simp
qed
lemma less_fun_trans:
assumes "less_fun f g" and "less_fun g h"
shows "less_fun f h"
proof (rule less_funI)
from \<open>less_fun f g\<close> obtain k1 where k1: "f k1 < g k1" "k' < k1 \<Longrightarrow> f k' = g k'" for k'
by (blast elim!: less_funE)
from \<open>less_fun g h\<close> obtain k2 where k2: "g k2 < h k2" "k' < k2 \<Longrightarrow> g k' = h k'" for k'
by (blast elim!: less_funE)
show "\<exists>k. f k < h k \<and> (\<forall>k'<k. f k' = h k')"
proof (cases k1 k2 rule: linorder_cases)
case equal with k1 k2 show ?thesis by (auto simp add: exI [of _ k2])
next
case less with k2 have "g k1 = h k1" "\<And>k'. k' < k1 \<Longrightarrow> g k' = h k'" by simp_all
with k1 show ?thesis by (auto intro: exI [of _ k1])
next
case greater with k1 have "f k2 = g k2" "\<And>k'. k' < k2 \<Longrightarrow> f k' = g k'" by simp_all
with k2 show ?thesis by (auto intro: exI [of _ k2])
qed
qed
lemma order_less_fun:
"class.order (\<lambda>f g. less_fun f g \<or> f = g) less_fun"
by (rule order_strictI) (auto intro: less_fun_trans intro!: less_fun_irrefl less_fun_asym)
lemma less_fun_trichotomy:
assumes "finite {k. f k \<noteq> g k}"
shows "less_fun f g \<or> f = g \<or> less_fun g f"
proof -
{ define K where "K = {k. f k \<noteq> g k}"
assume "f \<noteq> g"
then obtain k' where "f k' \<noteq> g k'" by auto
then have [simp]: "K \<noteq> {}" by (auto simp add: K_def)
with assms have [simp]: "finite K" by (simp add: K_def)
define q where "q = Min K"
then have "q \<in> K" and "\<And>k. k \<in> K \<Longrightarrow> k \<ge> q" by auto
then have "\<And>k. \<not> k \<ge> q \<Longrightarrow> k \<notin> K" by blast
then have *: "\<And>k. k < q \<Longrightarrow> f k = g k" by (simp add: K_def)
from \<open>q \<in> K\<close> have "f q \<noteq> g q" by (simp add: K_def)
then have "f q < g q \<or> f q > g q" by auto
with * have "less_fun f g \<or> less_fun g f"
by (auto intro!: less_funI)
} then show ?thesis by blast
qed
end